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Автор: Grafiati
Опубліковано: 8 червня 2024

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1

Montaldi, James, and Duco van Straten. "One-forms on singular curves and the topology of real curve singularities." Topology 29, no. 4 (1990): 501–10. http://dx.doi.org/10.1016/0040-9383(90)90018-f.

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2

Kleiman, Steven Lawrence, and Renato Vidal Martins. "The canonical model of a singular curve." Geometriae Dedicata 139, no. 1 (February 11, 2009): 139–66. http://dx.doi.org/10.1007/s10711-008-9331-4.

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3

Castañeda, Ángel Luis Muñoz. "On the moduli spaces of singular principal bundles on stable curves." Advances in Geometry 20, no. 4 (October 27, 2020): 573–84. http://dx.doi.org/10.1515/advgeom-2020-0003.

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AbstractWe prove the existence of a linearization for singular principal G-bundles not depending on the base curve. This allow us to construct the relative compact moduli space of δ-(semi)stable singular principal G-bundles over families of reduced projective and connected nodal curves, and to reduce the construction of the universal moduli space over 𝓜g to the construction of the universal moduli space of swamps.
4

Golla, Marco, and Laura Starkston. "The symplectic isotopy problem for rational cuspidal curves." Compositio Mathematica 158, no. 7 (July 2022): 1595–682. http://dx.doi.org/10.1112/s0010437x2200762x.

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We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology.
5

Menegon Neto, Aurélio. "Lê's polyhedron for line singularities." International Journal of Mathematics 25, no. 13 (December 2014): 1450114. http://dx.doi.org/10.1142/s0129167x14501146.

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We study the topology of line singularities, which are complex hypersurface germs with non-isolated singularity given by a smooth curve. We describe the degeneration of its Milnor fiber to the singular hypersurface by means of a vanishing polyhedron in the Milnor fiber. As a milestone, we also study the topology of the degeneration of a complex isolated singularity hypersurface under a nonlocal point of view.
6

Yang, Jieyin, Xiaohong Jia, and Dong-Ming Yan. "Topology Guaranteed B-Spline Surface/Surface Intersection." ACM Transactions on Graphics 42, no. 6 (December 5, 2023): 1–16. http://dx.doi.org/10.1145/3618349.

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The surface/surface intersection technique serves as one of the most fundamental functions in modern Computer Aided Design (CAD) systems. Despite the long research history and successful applications of surface intersection algorithms in various CAD industrial software, challenges still exist in balancing computational efficiency, accuracy, as well as topology correctness. Specifically, most practical intersection algorithms fail to guarantee the correct topology of the intersection curve(s) when two surfaces are in near-critical positions, which brings instability to CAD systems. Even in one of the most successfully used commercial geometry engines ACIS, such complicated intersection topology can still be a tough nut to crack. In this paper, we present a practical topology guaranteed algorithm for computing the intersection loci of two B-spline surfaces. Our algorithm well treats all types of common and complicated intersection topology with practical efficiency, including those intersections with multiple branches or cross singularities, contacts in several isolated singular points or highorder contacts along a curve, as well as intersections along boundary curves. We present representative examples of these hard topology situations that challenge not only the open-source geometry engine OCCT but also the commercial engine ACIS. We compare our algorithm in both efficiency and topology correctness on plenty of common and complicated models with the open-source intersection package in SISL, OCCT, and the commercial engine ACIS.
7

Nishimura, Takashi. "Normal forms for singularities of pedal curves produced by non-singular dual curve germs in S n." Geometriae Dedicata 133, no. 1 (January 30, 2008): 59–66. http://dx.doi.org/10.1007/s10711-008-9233-5.

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8

Fomin, Sergey, and Eugenii Shustin. "Expressive curves." Communications of the American Mathematical Society 3, no. 10 (August 28, 2023): 669–743. http://dx.doi.org/10.1090/cams/12.

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We initiate the study of a class of real plane algebraic curves which we call expressive. These are the curves whose defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of a curve. This concept can be viewed as a global version of the notion of a real morsification of an isolated plane curve singularity. We prove that a plane curve C C is expressive if (a) each irreducible component of C C can be parametrized by real polynomials (either ordinary or trigonometric), (b) all singular points of C C in the affine plane are ordinary hyperbolic nodes, and (c) the set of real points of C C in the affine plane is connected. Conversely, an expressive curve with real irreducible components must satisfy conditions (a)–(c), unless it exhibits some exotic behaviour at infinity. We describe several constructions that produce expressive curves, and discuss a large number of examples, including: arrangements of lines, parabolas, and circles; Chebyshev and Lissajous curves; hypotrochoids and epitrochoids; and much more.
9

Pinsky, Tali. "On the topology of the Lorenz system." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2205 (September 2017): 20170374. http://dx.doi.org/10.1098/rspa.2017.0374.

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We present a new paradigm for three-dimensional chaos, and specifically for the Lorenz equations. The main difficulty in these equations and for a generic flow in dimension 3 is the existence of singularities. We show how to use knot theory as a way to remove the singularities. Specifically, we claim: (i) for certain parameters, the Lorenz system has an invariant one-dimensional curve, which is a trefoil knot. The knot is a union of invariant manifolds of the singular points. (ii) The flow is topologically equivalent to an Anosov flow on the complement of this curve, and moreover to a geodesic flow. (iii) When varying the parameters, the system exhibits topological phase transitions, i.e. for special parameter values, it will be topologically equivalent to an Anosov flow on a knot complement. Different knots appear for different parameter values and each knot controls the dynamics at nearby parameters.
10

Guo, Feng, Gang Cheng, and Zunzhong Zhao. "Interior singularity analysis for a 2(3HUS+S) parallel manipulator with descending matrix rank method." International Journal of Advanced Robotic Systems 16, no. 1 (January 1, 2019): 172988141982684. http://dx.doi.org/10.1177/1729881419826841.

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Singularity analysis is one of the basic problems for parallel manipulators. When a manipulator moves in a singular configuration, the motion and transmission performance are poor. In certain serious cases, the normal operation could be damaged. Based on the topology structure and kinematics analysis of a 2(3HUS+S) parallel manipulator, the Jacobian matrices were established. Then, the singular locus surface was obtained by numerical simulation. In addition, the relationship between the motion path curve and the singular locus surface was analyzed. In this study, α, β, and γ are the attitude angles that describe the motion of moving platforms. There is a nonsingular attitude space in singular locus surfaces, and the singular locus surface is a single surface in a small attitude angle range. The nonsingular attitude space increases as the absolute value of γ increases, and singularity could be avoided when γ is large. Furthermore, the motion path curve passes through the singular locus surface two times, and the two intersection points are consistent with the positions where the motion dexterity is equal to zero. This study provides new insights on the singularity analysis of parallel manipulators, particularly for the structure parameter optimization of the nonsingular attitude space.
11

Bruno, Alexander D., and Alijon A. Azimov. "Parametric Expansions of an Algebraic Variety Near Its Singularities II." Axioms 13, no. 2 (February 4, 2024): 106. http://dx.doi.org/10.3390/axioms13020106.

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The paper is a continuation and completion of the paper Bruno, A.D.; Azimov, A.A. Parametric Expansions of an Algebraic Variety Near Its Singularities. Axioms 2023, 5, 469, where we calculated parametric expansions of the three-dimensional algebraic manifold Ω, which appeared in theoretical physics, near its 3 singular points and near its one line of singular points. For that we used algorithms of Nonlinear Analysis: extraction of truncated polynomials, using the Newton polyhedron, their power transformations and Formal Generalized Implicit Function Theorem. Here we calculate parametric expansions of the manifold Ω near its one more singular point, near two curves of singular points and near infinity. Here we use 3 new things: (1) computation in algebraic extension of the field of rational numbers, (2) expansions near a curve of singular points and (3) calculation of branches near infinity.
12

Zhang, Longjie. "On Curvature Flow with Driving Force Starting as Singular Initial Curve in the Plane." Journal of Geometric Analysis 30, no. 2 (December 1, 2017): 2036–91. http://dx.doi.org/10.1007/s12220-017-9937-6.

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13

Saad, M. Khalifa, H. S. Abdel-Aziz, and A. A. Abdel-Salam. "Evolutes of Fronts in de Sitter and Hyperbolic Spheres." International Journal of Analysis and Applications 20 (September 21, 2022): 47. http://dx.doi.org/10.28924/2291-8639-20-2022-47.

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The evolute of a regular curve is a classical object from the viewpoint of differential geometry. We study some types of curves such as framed curves, framed immersion curves, frontal curves and front curves in 2-dimensional de Sitter and hyperbolic spaces. Also, we investigate the evolutes and some of their properties of fronts at singular points under some conditions. Finally, some computational examples in support of our main results are given and plotted.
14

Mattei, J. F., J. C. Rebelo, and H. Reis. "Generic pseudogroups on and the topology of leaves." Compositio Mathematica 149, no. 8 (July 1, 2013): 1401–30. http://dx.doi.org/10.1112/s0010437x13007161.

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AbstractWe show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about $0\in \mathbb{C} $ is free in the sense of pseudogroups even if the class of conjugacy of the generators is fixed. This result has a number of consequences on the topology of leaves for a (singular) holomorphic foliation defined on a neighborhood of an invariant curve. In particular, in the classical and simplest case arising from local nilpotent foliations possessing a unique separatrix which is given by a cusp of the form $\{ {y}^{2} - {x}^{2n+ 1} = 0\} $, our results allow us to settle the problem of showing that a generic foliation possesses only countably many non-simply connected leaves.
15

Étienne GHYS and Christopher-Lloyd SIMON. "On the topology of a real analytic curve in the neighborhood of a singular point." Astérisque 415 (2020): 1–33. http://dx.doi.org/10.24033/ast.1097.

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16

Étienne GHYS and Christopher-Lloyd SIMON. "On the topology of a real analytic curve in the neighborhood of a singular point." Astérisque 415 (2020): 1–33. http://dx.doi.org/10.24033/ast.11097.

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17

Langer, Joel C., and David A. Singer. "Orthogonal Families of Bicircular Quartics, Quadratic Differentials, and Edwards Normal Form." Axioms 12, no. 9 (September 9, 2023): 870. http://dx.doi.org/10.3390/axioms12090870.

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Orthogonal families of bicircular quartics are naturally viewed as pairs of singular foliations of C^ by vertical and horizontal trajectories of a non-vanishing quadratic differential. Yet the identification of these trajectories with real quartics in CP2 is subtle. Here, we give an efficient, geometric argument in the course of updating the classical theory of confocal families in the modern language of quadratic differentials and the Edwards normal form for elliptic curves. In particular, we define a parameterized Edwards transformation, providing explicit birational equivalence between each curve in a confocal family and a fixed curve in normal form.
18

Xin, Li Li, and Ji Hui Liang. "Topology Analysis of Vibrating Compacting System with the Hysteretic Characteristics of Materials Considered." Advanced Materials Research 462 (February 2012): 93–97. http://dx.doi.org/10.4028/www.scientific.net/amr.462.93.

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Materials present elastic-plastic deformation under cyclical loading because of the influence of vibration friction, and restoring force and the displacement show hysteretic characteristics in compacting mechanism and materials system. To further study nonlinear dynamics of the vibrating compacting system, system model with symmetric hysteretic curve was established in this paper, and was verified by experiment. Bifurcation and chaos motion were studied in nonlinear vibrating compacting system by the combination of qualitative theory and numerical solution, and analysis results show that vibration have piecewise and singular characteristics in vibrating compacting system, and hysteretic loop parameters show significant influence on system and large effect on nonlinear dynamics of system.
19

CASALE, MALCOLM S. "EVALUATING GEOMETRIC SENSITIVITIES AT SINGULAR AND NON-SINGULAR POINTS OF VARIATIONAL CAD MODELS FOR DESIGN OPTIMIZATION." International Journal of Computational Geometry & Applications 06, no. 04 (December 1996): 443–60. http://dx.doi.org/10.1142/s0218195996000289.

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Much has been published on the topic of design optimization of mechanical components. Most of the research has concentrated on the finite element or boundary element part of the problem. Little effort has been applied to integrating design optimization with CAD. With the introduction of parametric and variational CAD, it is more desirable than ever to merge these technologies, i.e., to perform the analysis directly on the CAD geometry and to use the CAD parameters/dimensions as design variables. In this paper, one part of the problem is examined, the calculation of geometric sensitivities on variational CAD geometry. It is shown that for a well-conditioned set of constraint equations, the geometric sensitivities are easily obtained by a straight-forward application of the implicit function theorem. When the constraint equations become singular, the situation is more complex. The nature of singularities is explored, and a method, based on rational transformations that are common in algebraic curve tracing, is suggested to resolve singular points. It is shown that the geometric sensitivity is a natural by-product of the transformation. The paper concludes with some symbolic algebra software, a Mathematica package, that was found to be useful in the investigation.
20

Zhao, Jun Zhao, Wu Jun Chen, Gong Yi Fu, and Rui Xiong Li. "Numerical Simulation Algorithm for the Forming Process of Tensile Cable-Strut Structure." Advanced Materials Research 250-253 (May 2011): 1375–84. http://dx.doi.org/10.4028/www.scientific.net/amr.250-253.1375.

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Numerical simulation of forming process is an issue that finds the solution of a series of unstable equilibrium states of cable-strut tensile structures with specific unstressed length. This is a long well unsolved theoretic question, which mostly lies in the couple of mechanism and elastic deformation. On the basis of the unstressed length, the theoretic parabolic curve element was adopted to simulate the cable with elastic tension and deformation. The vector of unbalance nodal force was subsequently formulated with the explicit branch-node matrix of topology in force-density method. And the dynamical relaxation method was used to solve the nonlinear equation to avoid the singular stiffness of equation. Numerical examples indicated that the proposed algorithm could calculate the unstable equilibrium state and simulate the forming process with the unstressed length of the active cables changing, and has some theoretical significance.
21

Caratelli, Diego, and Paolo Emilio Ricci. "Logarithm of a Non-Singular Complex Matrix via the Dunford–Taylor Integral." Axioms 11, no. 2 (January 27, 2022): 51. http://dx.doi.org/10.3390/axioms11020051.

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Using the Dunford–Taylor integral and a representation formula for the resolvent of a non-singular complex matrix, we find the logarithm of a non-singular complex matrix applying the Cauchy’s residue theorem if the matrix eigenvalues are known or a circuit integral extended to a curve surrounding the spectrum. The logarithm function that can be found using this technique is essentially unique. To define a version of the logarithm with multiple values analogous to the one existing in the case of complex variables, we introduce a definition for the argument of a matrix, showing the possibility of finding equations similar to those of the scalar case. In the last section, numerical experiments performed by the first author, using the computer algebra program Mathematica©, confirm the effectiveness of this methodology. They include the logarithm of matrices of the fifth, sixth and seventh order.
22

Kitsiranuwat, Satanat, Apichat Suratanee, and Kitiporn Plaimas. "Integration of various protein similarities using random forest technique to infer augmented drug-protein matrix for enhancing drug-disease association prediction." Science Progress 105, no. 3 (July 2022): 003685042211092. http://dx.doi.org/10.1177/00368504221109215.

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Identifying new therapeutic indications for existing drugs is a major challenge in drug repositioning. Most computational drug repositioning methods focus on known targets. Analyzing multiple aspects of various protein associations provides an opportunity to discover underlying drug-associated proteins that can be used to improve the performance of the drug repositioning approaches. In this study, machine learning models were developed based on the similarities of diversified biological features, including protein interaction, topological network, sequence alignment, and biological function to predict protein pairs associating with the same drugs. The crucial set of features was identified, and the high performances of protein pair predictions were achieved with an area under the curve (AUC) value of more than 93%. Based on drug chemical structures, the drug similarity levels of the promising protein pairs were used to quantify the inferred drug-associated proteins. Furthermore, these proteins were employed to establish an augmented drug-protein matrix to enhance the efficiency of three existing drug repositioning techniques: a similarity constrained matrix factorization for the drug-disease associations (SCMFDD), an ensemble meta-paths and singular value decomposition (EMP-SVD) model, and a topology similarity and singular value decomposition (TS-SVD) technique. The results showed that the augmented matrix helped to improve the performance up to 4% more in comparison to the original matrix for SCMFDD and EMP-SVD, and about 1% more for TS-SVD. In summary, inferring new protein pairs related to the same drugs increase the opportunity to reveal missing drug-associated proteins that are important for drug development via the drug repositioning technique.
23

Piontkowski, Jens. "Topology of the compactified Jacobians of singular curves." Mathematische Zeitschrift 255, no. 1 (July 7, 2006): 195–226. http://dx.doi.org/10.1007/s00209-006-0021-3.

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24

Creutz, Paul. "Plateau’s problem for singular curves." Communications in Analysis and Geometry 30, no. 8 (2022): 1779–92. http://dx.doi.org/10.4310/cag.2022.v30.n8.a3.

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25

Cerbu, Alois, Steffen Marcus, Luke Peilen, Dhruv Ranganathan, and Andrew Salmon. "Topology of tropical moduli of weighted stable curves." Advances in Geometry 20, no. 4 (October 27, 2020): 445–62. http://dx.doi.org/10.1515/advgeom-2019-0034.

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AbstractThe moduli space Δg,w of tropical w-weighted stable curves of volume 1 is naturally identified with the dual complex of the divisor of singular curves in Hassett’s spaces of w-weighted stable curves. If at least two of the weights are 1, we prove that Δ0, w is homotopic to a wedge sum of spheres, possibly of varying dimensions. Under additional natural hypotheses on the weight vector, we establish explicit formulas for the Betti numbers of the spaces. We exhibit infinite families of weights for which the space Δ0,w is disconnected and for which the fundamental group of Δ0,w has torsion. In the latter case, the universal cover is shown to have a natural modular interpretation. This places the weighted variant of the space in stark contrast to the heavy/light cases studied previously by Vogtmann and Cavalieri–Hampe–Markwig–Ranganathan. Finally, we prove a structural result relating the spaces of weighted stable curves in genus 0 and 1, and leverage this to extend several of our genus 0 results to the spaces Δ1,w.
26

Ishikawa, Go-o. "Classifying singular Legendre curves by contactomorphisms." Journal of Geometry and Physics 52, no. 2 (October 2004): 113–26. http://dx.doi.org/10.1016/j.geomphys.2004.02.004.

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27

Unver, Demir. "Singular homology algorithm for MA-spaces." Thermal Science 23, Suppl. 6 (2019): 2139–47. http://dx.doi.org/10.2298/tsci190906403u.

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The work on digitizing subspaces of the 2-D Euclidean space with a certain digital approach is an important discipline in both digital geometry and topology. The present work considers Marcus-Wyse topological approach which was established for studying 2-D digital spaces, ?2. We introduce the digital singular homology groups of MA-spaces (M-topological space with an M-adjacency), and we compute singular homology groups of some certain MA-spaces, we give a formula for singular homology groups of 2-D simple closed MA-curves, and an algorithm for determining homology groups of an arbitrary MA-space.
28

Basu, Somnath, and Ritwik Mukherjee. "Enumeration of curves with one singular point." Journal of Geometry and Physics 104 (June 2016): 175–203. http://dx.doi.org/10.1016/j.geomphys.2016.02.008.

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29

Kannan, Siddarth, Shiyue Li, Stefano Serpente, and Claudia He Yun. "Topology of tropical moduli spaces of weighted stable curves in higher genus." Advances in Geometry 23, no. 3 (August 1, 2023): 305–14. http://dx.doi.org/10.1515/advgeom-2023-0009.

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Abstract We study the topology of moduli spaces of weighted stable tropical curves Δg ,w with fixed genus and unit volume. The space Δg ,w arises as the dual complex of the divisor of singular curves in Hassett’s moduli space M g ,w of weighted stable curves. When the genus is positive, we show that Δg ,w is simply connected for any choice of weight vector w. We also give a formula for the Euler characteristic of Δg ,w in terms of the combinatorics of the weight vector.
30

Gatto, Letterio. "Weight sequences versus gap sequences at singular points of Gorenstein curves." Geometriae Dedicata 54, no. 3 (March 1995): 267–300. http://dx.doi.org/10.1007/bf01265343.

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31

Neciosup Puican, Hernán. "Sobre la topología del complemento de una curva singular plana en la clasificación de foliaciones holomorfas singulares de codimensión uno." Pesquimat 26, no. 1 (June 30, 2023): 88–96. http://dx.doi.org/10.15381/pesquimat.v26i1.25068.

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En este artículo, estudiamos el papel del grupo fundamental del complemento de una curva plana afín en la clasificación analítica de foliaciones singulares de codimensión uno en (C3, 0). Nos enfocamos en obtener una representación adecuada del grupo fundamental del complemento de una curva plana afín particular, utilizando la monodromía de trenzas y el método de Zariski-Van Kampen. La imagen de este grupo, por la representación de holomonía de la foliación, es conocida como el grupo de holonomía de la foliación y la conjugación analítica de estos grupos equivale a la clasificación analítica de foliaciones holomorfas singulares cuspidales casi homogéneas de tipo admisible sobre (C3, 0) [6].
32

Bradlow, Steve, Lucas Branco, and Laura P. Schaposnik. "Orthogonal Higgs bundles with singular spectral curves." Communications in Analysis and Geometry 28, no. 8 (2020): 1895–931. http://dx.doi.org/10.4310/cag.2020.v28.n8.a6.

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33

Bradlow, Steve, Lucas Branco, and Laura P. Schaposnik. "Orthogonal Higgs bundles with singular spectral curves." Communications in Analysis and Geometry 28, no. 8 (2020): 1895–931. http://dx.doi.org/10.4310/cag.2020.v28.n8.a6.

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34

Spoar, Gary. "Multiplicities of singular points on arcs and curves of cyclic order four." Journal of Geometry 24, no. 1 (March 1985): 89–100. http://dx.doi.org/10.1007/bf01223536.

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35

Johnson, Drew, та Alexander Polishchuk. "Birational models of 𝓜2,2 arising as moduli of curves with nonspecial divisors". Advances in Geometry 21, № 1 (1 січня 2021): 23–43. http://dx.doi.org/10.1515/advgeom-2020-0026.

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Abstract We study birational projective models of 𝓜2,2 obtained from the moduli space of curves with nonspecial divisors. We describe geometrically which singular curves appear in these models and show that one of them is obtained by blowing down the Weierstrass divisor in the moduli stack of 𝓩-stable curves 𝓜 2,2(𝓩) defined by Smyth. As a corollary, we prove projectivity of the coarse moduli space M 2,2(𝓩).
36

ARTAL, E., J. CARMONA, J. I. COGOLLUDO, and HIRO-O. TOKUNAGA. "SEXTICS WITH SINGULAR POINTS IN SPECIAL POSITION." Journal of Knot Theory and Its Ramifications 10, no. 04 (June 2001): 547–78. http://dx.doi.org/10.1142/s0218216501001001.

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In this paper we show a Zariski pair of sextics which is not a degeneration of the original example given by Zariski. This is the first example of this kind known. The two curves of the pair have a trivial Alexander polynomial. The difference in the topology of their complements can only be detected via finer invariants or techniques. In our case the generic braid monodromies, the fundamental groups, the characteristic varieties and the existence of dihedral coverings of ℙ2 ramified along them can be used to distinguish the two sextics. Our intention is not only to use different methods and give a general description of them but also to bring together different perspectives of the same problem.
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Alcántara, Claudia R., and Rubí Pantaleón-Mondragón. "Foliations on $$\mathbb {CP}^2$$ with a unique singular point without invariant algebraic curves." Geometriae Dedicata 207, no. 1 (October 29, 2019): 193–200. http://dx.doi.org/10.1007/s10711-019-00492-8.

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38

Ballico, Edoardo, and Changho Keem. "On plane curves with several singular points with high multiplicity." Hiroshima Mathematical Journal 26, no. 1 (1996): 117–26. http://dx.doi.org/10.32917/hmj/1206127492.

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39

Melander, Mogens V., and Fazle Hussain. "Topological vortex dynamics in axisymmetric viscous flows." Journal of Fluid Mechanics 260 (February 10, 1994): 57–80. http://dx.doi.org/10.1017/s0022112094003435.

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The topology of vortex lines and surfaces is examined in incompressible viscous axisymmetric flows with swirl. We argue that the evolving topology of the vorticity field must be examined in terms of axisymmetric vortex surfaces rather than lines, because only the surfaces enjoy structural stability. The meridional cross-sections of these surfaces are the orbits of a dynamical system with the azimuthal circulation being a Hamiltonian H and with time as a bifurcation parameter μ. The dependence of H on μ is governed by the Navier–Stokes equations; their numerical solutions provide H. The level curves of H establish a time history for the motion of vortex surfaces, so that the circulation they contain remains constant. Equivalently, there exists a virtual velocity field in which the motion of the vortex surfaces is frozen almost everywhere; the exceptions occur at critical points in the phase portrait where the virtual velocity is singular. The separatrices emerging from saddle points partition the phase portrait into islands; each island corresponds to a structurally stable vortex structure. By using the flux of the meridional vorticity field, we obtain a precise definition of reconnection: the transfer of flux between islands. Local analysis near critical points shows that the virtual velocity (because of its singular behaviour) performs ‘cut-and-connect’ of vortex surfaces with the correct rate of circulation transfer - thereby validating the long-standing viscous ‘cut-and-connect’ scenario which implicitly assumes that vortex surfaces (and vortex lines) can be followed over a short period of time in a viscous fluid. Bifurcations in the phase portrait represent (contrary to reconnection) changes in the topology of the vorticity field, where islands spontaneously appear or disappear. Often such topology changes are catastrophic, because islands emerge or perish with finite circulation. These and other phenomena are illustrated by direct numerical simulations of vortex rings at a Reynolds number of 800.
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Chien, Mao-Ting, and Hiroshi Nakazato. "Singular points of the algebraic curves of symmetric hyperbolic forms." Linear Algebra and its Applications 470 (April 2015): 40–50. http://dx.doi.org/10.1016/j.laa.2014.02.006.

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41

Chien, Mao-Ting, and Hiroshi Nakazato. "Singular points of the algebraic curves associated to unitary bordering matrices." Linear Algebra and its Applications 513 (January 2017): 224–39. http://dx.doi.org/10.1016/j.laa.2016.10.018.

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42

Álvarez-Vizoso, J., Robert Arn, Michael Kirby, Chris Peterson, and Bruce Draper. "Geometry of curves in Rn from the local singular value decomposition." Linear Algebra and its Applications 571 (June 2019): 180–202. http://dx.doi.org/10.1016/j.laa.2019.02.006.

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43

Gómez, Tomás L. "Brill–Noether theory on singular curves and torsion-free sheaves on surfaces." Communications in Analysis and Geometry 9, no. 4 (2001): 725–56. http://dx.doi.org/10.4310/cag.2001.v9.n4.a3.

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44

Lo Giudice, Alessio, and Andrea Pustetto. "A compactification of the moduli space of principal Higgs bundles over singular curves." Journal of Geometry and Physics 110 (December 2016): 328–42. http://dx.doi.org/10.1016/j.geomphys.2016.08.007.

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45

Malchiodi, A. "Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains." GAFA Geometric And Functional Analysis 15, no. 6 (December 2005): 1162–222. http://dx.doi.org/10.1007/s00039-005-0542-7.

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46

Benini, Anna Miriam, and Lasse Rempe. "A landing theorem for entire functions with bounded post-singular sets." Geometric and Functional Analysis 30, no. 6 (November 20, 2020): 1465–530. http://dx.doi.org/10.1007/s00039-020-00551-3.

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AbstractThe Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial f with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue of this theorem for an entire function f with bounded postsingular set. If f has finite order of growth, then it is known that the escaping set I(f) contains certain curves called periodic hairs; we show that every periodic hair lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic hair. For a postsingularly bounded entire function f of infinite order, such hairs may not exist. Therefore we introduce certain dynamically natural connected subsets of I(f), called dreadlocks. We show that every periodic dreadlock lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic dreadlock. More generally, we prove that every point of a hyperbolic set is the landing point of a dreadlock.
47

Greenblatt, Michael. "An Analogue to a Theorem of Fefferman and Phong for Averaging Operators Along Curves with Singular Fractional Integral Kernel." Geometric and Functional Analysis 17, no. 4 (September 17, 2007): 1106–38. http://dx.doi.org/10.1007/s00039-007-0622-y.

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48

Bruno, Alexander D., and Alijon A. Azimov. "Parametric Expansions of an Algebraic Variety near Its Singularities." Axioms 12, no. 5 (May 13, 2023): 469. http://dx.doi.org/10.3390/axioms12050469.

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Presently, there is a method based on Power Geometry that allows one to find asymptotic forms and asymptotic expansions of solutions to different kinds of non-linear equations near their singularities. The method contains three algorithms: (1) Reducing the equation to its normal form, (2) separating truncated equations, and (3) power transformations of coordinates. Here, we describe the method for the simplest case, a single algebraic equation, and apply it to an algebraic variety, as described by an algebraic equation of order 12 in three variables. The variety was considered in study of Einstein’s metrics and has several singular points and singular curves. Near some of them, we compute a local parametric expansion of the variety.
49

Jung, Junehyuk, and Steve Zelditch. "Number of nodal domains and singular points of eigenfunctions of negatively curved surfaces with an isometric involution." Journal of Differential Geometry 102, no. 1 (January 2016): 37–66. http://dx.doi.org/10.4310/jdg/1452002877.

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50

Shi, Feng, and Kang-Jia Wang. "Various Solitons and Other Wave Solutions to the (2+1)-Dimensional Heisenberg Ferromagnetic Spin Chain Dynamical Model." Axioms 12, no. 4 (April 3, 2023): 354. http://dx.doi.org/10.3390/axioms12040354.

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This paper outlines a study into the exact solutions of the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation that is used to illustrate the ferromagnetic materials of magnetic ordering by applying two recent techniques, namely, the Sardar-subequation method and extended rational sine–cosine and sinh–cosh methods. Abundant exact solutions such as the bright soliton, dark soliton, combined bright–dark soliton, singular soliton and other periodic wave solutions expressed by the generalized trigonometric, generalized hyperbolic, trigonometric and hyperbolic functions are obtained. The numerical results are illustrated in the form of 3D plots, 2D contours and 2D curves by choosing proper parametric values to interpret the physical behavior of the model. The obtained results in this work are expected to provide a rich platform for constructing the soliton solutions of PDEs in physics.

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